Dump the grated potatoes and onions in the center of a large, clean kitchen towel. Gather the sides of the towel on top of the potatoes and onions and squeeze the mixture tightly—as tightly as you can—over the kitchen sink to remove any excess liquid. Transfer the grated, drained potatoes and onions to a large bowl and stir in the Cheddar, flour, eggs, salt, and pepper. Toss to combine.
Heat 1/4 cup oil in a 10-inch sauté pan or cast-iron skillet over medium heat. When the oil is hot, add as many 1/4-cupfuls batter as will comfortably fit, flattening each circle a bit with the bottom of the 1/4-cup measuring cup or a spatula. (Don’t try to fry more than 5 latkes at a time.)
Cook until the first side is golden brown and crisp, 2 1/2 to 3 1/2 minutes. Flip and cook the other side until also golden brown and crisp, about another 3 1/2 minutes. Transfer the finished latkes to a paper towel–lined baking sheet.
Repeat with the remaining latke batter, not letting the pan go dry (you’ll probably only need another 1 tablespoon oil). You should have about 12 latkes.
Serve the latkes with the applesauce and sour cream, if desired.